2013中国女子数学奥林匹克试题及其解答(3)

1.证明（张云华）

1.证明（mavropnevma ）

Let

and

,

. The region

and

is the triangle . But

is

for

(equality ). ,

for

,

, thus the triangles of largest possible area are

(equality

for disallowed).is for

EDIT. Referring to the next post - it is interesting to figure out the envelope of the

lines

is the

parabola

functions for a fixed ,

with

triangular area

, but in some way it is irrelevant, since the problem fixed on the parabola for

some . , and the

being delimited by the tangent to that parabola at

1.证明（kunny

）

parabola

Which touches

The domain

points at is envelope of the lines

. , thus the

of the family of the lines swept by is shown by the shaded region, excluded two

.

Edit:I was misreading the context problem, thank you for pointing put it, mavropnevma.